504 



NATURE 



[August 29, 19 18 



LETTERS TO THE EDITOR. 



[The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.] 



Production of Medusoid Forms from Gels. 



The reference lo the phenomena- of ordinary drops 

 in Prof. D'Arcy W. Thompson's letter on "Medusoid 

 Bells" iii Nature of August 8 has suggested to me 

 the possibility of obtaining permanent imitations of 

 such forms as he describes by producing drops of 

 gelatin in a suitable medium. The latter must be 

 one of the solutions which harden gelatin, must have 

 a specific gravity very near that of the gelatin sol 

 at the temperature at which it is used, and must 

 possess an appreciable interfacial tension against the 

 sol. I have found that a solution of aluminium 

 sulphate can be made which fulfils all these con- 

 ditions. 



If 20 per cent, gelatin sol, which may be coloured 

 with any convenient dye, is dropped into such a -solu- 

 tion from a tube about 4 mm. diameter, with its orifice 

 from 2 to 8 mm. above the surface, rather interesting 

 forms are obtained. The specimens do not lend them- 

 selves very well to photographic reproduction, but I 

 have drawn diagrammatically three typical cases. In 

 all instances the crenated or stellate portion rests on 

 the surface. With a 10 per cent, gelatin sol per- 

 manent vortex rings can be obtained, as well ,as discs 

 with a thickened rim, rings with a cylindrical fringe, 

 etc. 



son FACE OF 



To approach more nearly to the conditions of the 

 budding organism, it would be necessary to discharge 

 the drops below the surface of the liquid. This pro- 

 cedure entails some experimental difficulties, which, 

 however, I hope to overcome. The forms §0 far pro- 

 duced do not show to me any evidence of vibration, 

 but appear to be completely explicable by the effects 

 of surface and interfacial tension and of the removal 

 of water from the gel. Further experiments mav 

 show such evidence, and it would be very interesting 

 if they could furnish support for so attractive a hypo- 

 thesis in view of its two prima facie diflficulties : the 

 origin and the persistence of vibration in a medium 

 with the peculiar elastic properties of dilute gols. Per- 

 haps the results described may induce others, more 

 competent than I am to interpret their biological and 

 morphological aspects, to make such experiments ; the 

 conditions may be varied in a great number of ways 

 which will readily suggest themselves to anyone 

 familiar with the properties of gelatin. 



Emil Hatschek. 



10 Nottingham Mansions, Nottingham 

 Stree'., W.i, August 16. 



NO. 2548, VOL. lOl] 



be 

 so 

 with the proper 



the volume 



Formulae for Tetrahedron. 



Perhaps some readers of Nature may be able to tell 

 me whether the following results are new : — 



Let ABCD be any tetrahedron; BC = a, DA = a', find 

 so for the other edges; (BC) = dihedral angle of which 

 BC is an edge, and so on ; {aa') the angle between 

 BC, AD, and so on ; o, /S, 7, 8 the areas of the faces 

 opposite .'\, B, C, D respectively. Then we have 

 identically 



aa cos {aa' ) -\- bb' cos {bb') + cc' cos{ic') = o (i) 



aa cos {aa') cos ( bC) cos (AD) + bb' cos (bb') cos (C A) cos (BD) 



+ cf' cos {a') cos ( AB) cos (CD) = o (ii) 



It is a known theorem, due to Steiner, that the four 

 altitudes of ABCD are generators of the same hyper- 

 boloid. With the help of (i) and (ii) I have found 

 that, taking ABCD as the tetrahedron of reference, 

 the equation of Steiner's hyperboloid is, in volume co- 

 ordinates, 



aa' cos {aa') \a5yz cos ( HC) + Byx/ cos ( AD)| 



+ bb' cos {bb')'fiS=x cos (CA) + yayi cos ( BD); 



' +cc' COS {<:c')\ySx}' cos {AB) + aj82^ cos (CD)| =0. 



In these formulae certain conventions have to 

 made in the definitions of the angles (aa'), etc. 

 as to make the cosines come out 

 signs. 



Another interesting result is that if \ 

 of the tetrahedron, 



4aByn jcos { AB) cos (CD) - cos (CA) cos ( BD)} 

 — g\-aa' cos {aa') 

 with two other identities derived from this by inter- 

 change of letters. 



All the formulae can be translated .into vector iden- 

 tities; thus i'aa' cos (aa') = o corresponds to the 

 quaternion identity 



S |(y8 - 7)0 + (7 - a)j8 + (a - /3)7} =0, 

 but the others do not seem to ma to be so easily 

 derivable. G. B. Mathews. 



7 Menai View, Bangor, August 17. 



Rotating Discs. 



A note in Nature, August 22, p. 491, referring to a 

 recent article by Mr. H. Haerle, says: "The problem 

 of ascertaining the distribution and magnitudes of the 

 stresses in a revolving disc b}- means of mathematical 

 formulas is tedious and complicated. With the excep- 

 tion of the cases of discs of constant thickness and 

 constant strength, for which definite integrals can be 

 found, the analytical solution involves highly complex 

 equations, and the ultimate result is doubtful." May 

 I point out that the ordinary approximate solution for 

 the rotating circular disc of uniform thickness, whether 

 complete or holed, involves only simple powers of the 

 radius vector? The corresponding solution for the thin 

 elliptical disc involves expressions of an equally ele- 

 mentary type, though naturally longer. But in addi- 

 tion we have possessed for more than twenty years 

 (see Proc. Roy. Soc, vol. Iviii., p. 39) a complete 

 solution for an ellipsoid of any shape rotating about 

 a principal axis. This involves only simple powers of 

 the variables x, y, g, and it applies, of course, to discs 

 of very varied shapes. All the ordinary elastic solid equa- 

 tions, whether internal or external, are exactly satisfied 

 in this case. Thus the uncertainties are only those 

 inevitable through the difference between the 

 ideal elastic solid problem and its realisation in 

 practice. C Chree. 



August 26. 



